analysis of focused laser differential...

Analysis of focused laser differentialinterferometryB. E. SCHMIDT* AND J. E. SHEPHERD

California Institute of Technology, 1200 E. California Blvd., Pasadena, California 91125, USA*Corresponding author: [email protected]

Received 10 July 2015; revised 31 August 2015; accepted 2 September 2015; posted 3 September 2015 (Doc. ID 245742);published 30 September 2015

A computational method for predicting the output of a focused laser differential interferometer (FLDI) given anarbitrary density field is presented. The method is verified against analytical predictions and experimental data.The FLDI simulation software is applied to the problem of measuring Mack-mode wave packets in a hyperve-locity boundary layer on a 5° half-angle cone. The software is shown to complement experiments by providing thenecessary information to allow quantitative density fluctuation magnitudes to be extracted from experimentalmeasurements. © 2015 Optical Society of America

OCIS codes: (120.3180) Interferometry; (000.2170) Equipment and techniques; (000.4430) Numerical approximation and analysis;

(140.0140) Lasers and laser optics.

http://dx.doi.org/10.1364/AO.54.008459

1. INTRODUCTION

Focused laser differential interferometry is a promising tech-nique for measuring density disturbances in supersonic andhypersonic flows, particularly in measuring boundary-layer in-stabilities. Focused laser differential interferometry is a subset oflaser differential interferometry, a field with many examples inthe literature, e.g., [1–3] and others. Texts that describe thegeneral principles of interferometry and describe several typesof interferometers are abundant, e.g., [4]. The focused laser dif-ferential interferometer (FLDI) was first described by Smeets[5] where it was used to measure density fluctuations in windtunnel flows and turbulent jets in a desktop-type experiment.The technique was limited in its usefulness at that time becauseof limitations on photodetectors and data acquisition systems aswell as the availability of suitable lasers. Parziale revitalized thetechnique in 2013 [6–8] to measure instabilities in a hyperve-locity boundary layer on a slender body in the T5 hypervelocitytunnel at Caltech and to make measurements of the free streamenvironment in T5 [9]. This paper will develop some generalresults for the FLDI but will concentrate on the application ofmeasuring second-mode (Mack) waves in hypersonic boundarylayers [10].

The FLDI is a very attractive instrument for making suchmeasurements for several reasons. It has high frequency re-sponse of greater than 10 MHz, spatial resolution of the orderof hundreds of microns in the streamwise direction, and a highsignal-to-noise ratio. Additionally, because of the focusing abil-ity of the FLDI, it rejects much of the unwanted signal awayfrom the flow feature of interest near the instrument’s best

focus. A key advantage then is that for many flows the FLDIis largely immune to large-amplitude density disturbances cre-ated by the shear layers of a wind tunnel with a free jet insidethe test section. Preliminary qualitative evidence of this prop-erty of the FLDI has been observed in experiments involvingtranslating a small turbulent jet, e.g., Section 3.2.3 of [11]. Theeffect was examined in detail by Fulghum in Section 3.10.2 of[12] and is also studied in this paper.

As more researchers use the technique, it is critical to betterunderstand how the FLDI signal is produced and how to prop-erly analyze experimental results to extract meaningful quanti-tative information about the fluctuating density field in theflow. Fulghum presents a very thorough description of theFLDI technique from an aero-optical point of view and derivessystem transfer functions for the instrument for a few simpleflow geometries [12]. This paper presents a computationalmethod for simulating the response of the FLDI to arbitrarydensity fields in order to determine the sensitivity of the instru-ment to more complicated flows with a special emphasis onmeasurements in hypersonic boundary layers.

2. FLDI THEORY

The essential operating principles of the FLDI are presentedhere; for a more complete explanation, the reader is referredto Section 3.6 of the Ph.D. thesis by Fulghum [12]. The FLDIis a nonimaging shearing interferometer. A sketch of the instru-ment layout is shown in Fig. 1. The linearly polarized laserbeam is expanded and sheared by a prism by a small angle σwhich is placed at the focal point of a converging lens. This fixes

Research Article Vol. 54, No. 28 / October 1 2015 / Applied Optics 8459

1559-128X/15/288459-14$15/0$15.00 © 2015 Optical Society of America

http://dx.doi.org/10.1364/AO.54.008459

the shear distance between the beams to Δx. The two beamshave mutually orthogonal polarization. Wollaston prisms arethe most common choice of prism in the literature, butFulghum demonstrates great success with Sanderson prisms[13], where the divergence angle can be adjusted and theaperture can be larger so as not to truncate the expanded beam.Sanderson prisms are also generally less expensive thanWollaston prisms with small divergence angles. The choice ofprism does not change the fundamental characteristics of theinterferometer [14] or impact the analysis presented here.An illustration of the operation of a prism is shown in Fig. 2.

After the prism, the focusing lens brings the beams to a sharpfocus. The system is symmetric about the focus so that the beamscan be recombined by means of a second polarizer and the in-terference signal is measured by a change in intensity on a photo-detector. Inhomogeneities are spatially filtered by the beams,with a much stronger filtering effect where the beam diameteris large, which makes the instrument most sensitive near thepoint of best focus and least sensitive close to thefocusing lenses on either side of the focus, which in a free-jetwind tunnel would be close to the turbulent shear layers atthe edges of the test flow. This spatial filtering effect allowsthe FLDI to “see through” the strong turbulence at the edgesof a wind tunnel flow and measure density fluctuations of muchlower intensity in the region of interest in the core of the tunnel.

As an interferometer, the FLDI is sensitive to phasedifferences between the two beams of the instrument.Equations describing the interference of two superimposedwaves are derived in Section 7.2 of [15]. The equations in this

section follow directly by considering a set of rays that are in-tegrated over a detector. A phase difference is created by achange in the index of refraction of a transparent medium alongthe paths of two rays according to

Δϕ � 2π

λ

�ZD�ξ;η�

s1n�x1�ds1 −

ZD�ξ;η�

s2n�x2�ds2

�: (1)

Here, n is the index of refraction field through which the rayspass, the vector xi represents the ray path parametrized by si,i.e., xi � �x�si�; y�si�; z�si��, D�ξ; η� is the point on the detec-tor where beams 1 and 2 terminate, and λ is the wavelength ofthe laser used. ξ and η are the coordinates on the detector face.Note that both rays terminate at the same point on the detector.Corresponding rays are separated in the test region byΔx in thex-direction, x1 � x2 � Δxx. If the rays are interfered in aninfinite fringe configuration, as they are in the FLDI, the in-tensity of the interfered ray at point �ξ; η� on the photodetectoris given by

I�ξ; η� � I1�ξ; η� � I 2�ξ; η�� 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI 1�ξ; η�I 2�ξ; η�

pcos�Δϕ�ξ; η��: (2)

If we assume that the two rays have the same initial intensity,I 1 � I 2 � I0

2, then Eq. (2) simplifies to

I�ξ; η�I 0�ξ; η�

� 1� cos�Δϕ�ξ; η��; (3)

where I 0�ξ; η� is the normalized intensity profile of the beam.In practice, we adjust the instrument to the middle of an in-terference fringe such that there is a constant phase shift of−π∕2 between the two beams so that Eq. (3) can be linearizedfor Δϕ ≪ 1 as

I�ξ; η�I0�ξ; η�

� 1� sin�Δϕ�ξ; η�� ≈ 1� Δϕ�ξ; η�: (4)

The signal output by the detector ΔΦ is proportional to theintegral of Eq. (4) over the detector face D which gives the totalweighted average phase change ΔΦ as

ΔΦ �Z Z

D�I�ξ; η� − I0�ξ; η��dξdη

�Z Z

DI 0�ξ; η�Δϕ�ξ; η�dξdη: (5)

Or, substituting Eq. (1),

ΔΦ �Z Z

D�I�ξ; η� − I 0�ξ; η��dξdη

� 2π

λ

Z ZDI 0�ξ; η�

�ZD�ξ;η�

s1n�x1�ds1

−

ZD�ξ;η�

s2n�x2�ds2

�dξdη: (6)

Finally, the index of refraction n in a gas is related to the densityof the gas by the Gladstone–Dale relation:

n � K ρ� 1: (7)

This allows the output of the FLDI to be related to the densityfield of the gas being probed.

The photodetector converts the total intensity to a voltage.Large phase changes (ΔΦ > π∕2) cause phase ambiguity to

Fig. 1. Schematic of an FLDI setup. The two beams are shown asblue and green. Regions where the beams overlap are shown as striped.The coordinate system shown will be the one used throughout thispaper.

Fig. 2. Illustration of a prism (here, a Wollaston prism). The inci-dent beam of arbitrary polarization is split into two beams by an angleσ, and the two beams at the exit have mutually orthogonal polariza-tion. The ordinary ray is linearly polarized in the direction of beamseparation and the extraordinary ray is polarized 90° from the directionof separation.

8460 Vol. 54, No. 28 / October 1 2015 / Applied Optics Research Article

occur, as the interference wraps over several periods of lightwaves. Therefore, it is best to keep the phase change smallenough such that the sine function can be linearized. Forthe FLDI, it is most useful to interpret the output not as a phasechange ΔΦ between the two closely spaced beams but rather asa finite-difference approximation to the first derivative of thephase change, ΔΦ∕Δx. For small values of Δx, this approxi-mates the first derivative of phase change in the direction ofbeam separation. Smaller values of Δx result in more accurateapproximations of the derivative and therefore increased fre-quency response, but smaller beam separations result in lowersignal magnitudes overall, which becomes an issue in practice asthe electronic noise floor is approached.

The first prism (Prism 1 in Fig. 1) not only splits the lightbeams, but the two beams exit the prism polarized orthogonallyto one another. Therefore, in order to compute the response ofthe instrument at the photodetector after the beams have beenrecombined, it is necessary to consider the state of polarizationof the light along the beam paths and perform an analysis likethe one in Section 3.6.1 of [12]. However, if the light is po-larized at 45° relative to the separation angle of the prism beforeentering the first prism, the equations governing the polariza-tion state simplify considerably. This is because each beam leav-ing the prism will have equal amplitude and the beams can berecombined and mixed on the detector side without explicitlyusing Jones vectors to combine the electric fields as long as thepolarization is not rotated by the optical system. This is theconfiguration used by Parziale [11].

3. COMPUTATIONAL METHOD

As analytically determining the response of the FLDI instru-ment for a given density field is extremely difficult for allbut the simplest flow geometries, a computational model ofthe FLDI is developed to numerically evaluate Eq. (6) for agiven arbitrary density field that can vary in space and timeand simulate the FLDI output. The software described in thissection is referred to as the FLDI software throughout thispaper. The software replicates the FLDI configuration used atCaltech [6] but can be modified to suit the dimensions of anyFLDI setup. Dimensions are given in Table 1. The general pro-cedure followed by the software is to first compute the regiontraversed by the FLDI beams and then to discretize the domainas described in this section. Finally, the integral in Eq. (6) isevaluated numerically along the beam paths.

The beams are assumed to have equal Gaussian intensitydistributions I0�ξ; η� and the beams are assumed to propagateaccording to Gaussian beam propagation. Assuming Gaussianpropagation means that the angle of paraxial rays andhigher-order terms can be neglected from the full electromag-netic wave propagation equations. This is a good approxima-tion as long as all the rays form a sufficiently small angle with

the primary beam axis such that the small-angle approximationcan be invoked. The validity of this assumption is examinedlater in this section. For a more detailed discussion of theapproximations involved in assuming Gaussian beam propaga-tion, see Chapter 4 of Born and Wolf [15].

The beam separation Δx is calculated by simple trigonom-etry to be

Δx � 2f tanσ

2� 174.5 μm: (8)

This calculation is confirmed to be accurate by photographingthe beams of the physical FLDI setup at Caltech near the bestfocus with a CCD camera and neutral density filters to preventsaturation.

Equations (9)–(11) can be found in Section 14.5 of [16].The beam waist radius at the best focus w0 is computed forGaussian beams using Eq. (9), which is found by substitutionfor the divergence angle of a Gaussian beam as

w0 ≈λ

πθd≈

2λdπD4σ

: (9)

For a diffraction-limited beam, this corresponds to a spot size ofabout 7 μm. The 1∕e2 radius of the beam as a function of z, thecoordinate along the beam path, is given by

w�z� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw20

�1�

�λzπw2

0

�2�s; (10)

where z � 0 at the beam waist.Figure 3 shows computed beam widths near the best focus

of the FLDI. For simplicity when calculating the beam profile,polar-cylindrical coordinates are used with r and θ orthogonalto z and r2 � x2 � y2. The normalized beam intensity crosssection at a point in z is then

I0�r� �2

w2�z�π exp

�−2r2

w2�z�

�: (11)

The computational domain encompassing the beams be-tween the focusing lenses is discretized into a uniform gridof 10,300 points along the beam paths, corresponding to adimensional step size of 100 μm which is found to be sufficientenough that the computation is not affected by the step size.Convergence is shown below in Section 4.A. The beam cross

Table 1. Optical Parameters for Simulated FLDI

Divergence angle of prisms (σ) 2 arc min1∕e2 beam diameter at focusing lens (D4σ) 48 mmFocal length of focusing lenses (f ) 300 mmDistance from focusing lens to focus (d ) 515 mmλ 532 nm

z [mm]-20 -10 0

x [m

m]

-1.5

-1

-0.5

0

0.5

1

1.5

Fig. 3. Computed beam widths (out to 1∕e2) within 30 mm of thebest focus. One beam is outlined in red and the other in blue. Thewidth of the beams at the waist is too small to see on this scale.


section is divided into a polar grid with r nondimensionalizedby w�z�, the local 1∕e2 beam radius. Therefore, each point�r0; θ0� on the polar grid at a point z1 is on the same ray asthe point �r0; θ0� at any other location in z. In this way thesoftware can be considered to be performing geometric ray trac-ing, except the eikonal equation is not used to evaluate ray de-flections due to the density field. Density perturbations areapproximated as small enough (in magnitude and in extentalong the ray path) that they only cause a change to the phaseof each individual ray but do not cause the rays to refract sig-nificantly. The beam profiles and rays are calculated assuming azero-disturbance field first, and then the total change in phaseof the beams is calculated using an input density field.Alternatively, the method can be thought of as computing apixelwise phase change for each beam on the face of the detec-tor, where each grid point at a cross section in z is a pixel. Such amethod is shown to produce accurate results compared to theparabolic beam method developed by White and the Rayleigh–Sommerfeld equation as long as the beam is not analyzed closeto an aperture [17]. Because the polar grid is normalized by thelocal beam radius, integration occurs along each individual raypath instead of along the z-axis.

The polar cross-section grid extends to r∕w � r � 2, whichcontains 99.99% of the beam energy. We can now consider ifassuming Gaussian beam propagation is accurate. The maxi-mum angle formed by a beam in the domain will be the angleformed by a beam at the outer edge of the grid. Using Eqs. (9)and (10) and Table 1, the maximum ray angle is calculated tobe 5.32° or 92.9 mrad. The small-angle approximation for thisangle gives an error of 0.14%, hence assuming Gaussian beampropagation is clearly justified. The grid has 300 equally spacedpoints in the θ-direction, and grid points are chosen in ther-direction such that each cell has an aspect ratio as close to1 as possible. Points are computed starting at r � 2 inwardto a specified limiting radius r0, which is chosen to be0.001, resulting in 363 points in r. Each point is computedusing the previous point according to

rk � rk−1

�2 − δθ

2� δθ

�; (12)

where δθ is the step size in the θ-direction. The grid also con-tains one point in the center at r � 0, bringing the total num-ber of points at each cross-sectional grid to 108,901. The grid isshown in Fig. 4. The resolution was determined by limiting theerror of numerically integrating a Gaussian using the trapezoi-dal rule on the grid to less than 1%.

The simulated FLDI response to an input density fieldρ�x; y; z� is computed by numerically evaluating Eq. (6).The integral in z is calculated using Simpson’s rule and thetwo-dimensional integral over the face of the beam is calculatedusing trapezoidal integration.

4. SOFTWARE VERIFICATION

A. System Transfer FunctionsIt is possible to analytically derive an overall system transferfunction H as a function of wavenumber for simple densitydisturbance fields. This is performed in detail in [12], the es-sence of which is summarized in this paper. Here, H is definedas the ratio of the output of the instrument to the actual firstderivative of the phase field as

H ≡

�ΔΦΔx

�meas:

dΦdx

: (13)

H for the FLDI is the convolution of two filters: one resultingfrom the finite beam separation approximating a derivative, andthe other resulting from the Gaussian intensity distribution ofthe beams. In wavenumber (k) domain, these filters are simplymultiplied together to give the overall H �k� for the system.Here, k is the wavenumber of the density disturbance fieldand not the wavenumber of the laser. In general, H �k� willbe different for every density field geometry in �x; y; z�-space.One simple field geometry that can be analyzed analytically is asinusoidal disturbance in x that is uniform in y and infinitesi-mally thin in z at z � 0, i.e., n 0 � A sin�kx�δ�z�, where A issome arbitrary disturbance amplitude and δ is the Dirac deltafunction. The transfer function from the Gaussian intensitydistribution of the beam Hw�k� can be derived fromEqs. (6), (11), and (13). We consider a detector over all spacewith the detector coordinates ξ and η aligned with Cartesiancoordinates x and y and take the limit as the beam separationΔx approaches zero:

Hw�k�

� 1ddx �sin�kx��x�0

limΔx→0

�1

Δx

Z∞Z−∞

I 0�x; y�

×�Z

D�x;y�

s1sin�kx�δ�z�ds1 −

ZD�x;y�

s2sin�kx�δ�z�ds2

�dxdy

�� 1

ddx �sin�kx��x�0

Z∞Z−∞

I 0�x; y�d

dx�sin�kx��dxdy

� 1

k

Z∞Z−∞

2

w2πexp

�−2�x2 � y2�

w2

�k cos�kx�dxdy

� exp

�−w2k2

8

�: (14)

At z � 0, this is simply

x/w-2 -1 0 1 2

y/w

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Fig. 4. Polar grid cross section nondimensionalized by the localbeam waist size.


Hw;0�k� � exp

�−w20k

2

8

�: (15)

Equation (14) reveals how the FLDI rejects an unwanted signalaway from the best focus. The FLDI rapidly attenuates disturb-ances with wavelengths sufficiently smaller than the local beamdiameter where the product of w and k is large. w is approx-imately linear in z away from the best focus, so a disturbancewith a given wavenumber is attenuated with Gaussian decay asit moves away from the focus.

The FLDI software can compute the response for a singlebeam probing k cos�kx� at a beam cross section at z � 0 overrelevant wavenumbers for disturbances in supersonic andhypersonic flows, and this is compared with the analytical resultof Eq. (15) in Fig. 5. The computed response curve is nearlyidentical to the analytical result. The response curve is flat witha value of 1 for low wavenumbers (long wavelength) with asharp exponential roll-off beginning at about k � 100∕mm,which is a disturbance wavelength of 63 μm, much smaller thanmost relevant waves in supersonic flows. From Eq. (14) it isapparent that the roll-off point occurs at lower wavenumbersfor larger beam diameters, i.e., further from the best focus. Thisis what makes the FLDI immune to density disturbances awayfrom the focus, such as the turbulent shear layers at the edges ofa free-jet supersonic wind tunnel.

Note, however, that since the FLDI is actually measuring afinite difference approximation to the first derivative of densityalong the direction of beam separation, the magnitude of theraw signal will be smallest where that derivative is smallest, i.e.,at low wavenumbers. Therefore, the effect of the electronicnoise floor will become significant for low wavenumber dis-turbances. The issue of the noise floor will be discussed laterin this section.

Hw�k� can also be computed analytically for a disturbancefield that is uniform in z but has a finite width 2L, i.e.,n 0 � A sin�kx��U �z � L� − U �z − L��, where U is theHeaviside step function. This is a more physically meaningfultransfer function than Eq. (15). Hw�k� is simply the integral ofEq. (14) from −L to L in z divided by 2L:

Hw�k� �1

2L

ZL

−Lexp

�−w20k

2

8

�1�

�λzπw2

0

�2��

dz

� πw0

ffiffiffiffiffi2π

p

kLλexp

�−w20k

2

8

�erf

�kLλ

2ffiffiffi2

pπw0

�: (16)

Equation (16) is plotted for various values of L, the densitydisturbance half-width in z, in Fig. 6. As L increases, roll-offbegins at lower values of k because the instrument is integratingover portions of the beam where the diameter is larger, thusfiltering high wavenumber disturbances according to Eq. (14).The error function in Eq. (16) introduces a k−1 roll-off thatextends until the Gaussian decay cuts in at a wavenumberof about 103∕mm.

Figures 7 and 8 compare the output of the FLDI softwarewith the analytical result from Eq. (16) for two values of L: 10and 30 mm. Excellent agreement is again observed betweenthe analytical and computed transfer functions, except at highwavenumbers where numerical errors manifest away from thebest focus where the beam is larger and the cross-sectional gridis therefore coarser with respect to the high wavenumberdisturbances.

In addition to the filtering effect due to the changing beamsize, there is a second filter due to the beams being separated bya finite distance. Hs�k�, the transfer function based on beamseparation, is calculated by computing the response of theFLDI by approximating the FLDI as two point-detectorsseparated by Δx, again compared to the ideal case of the truederivative of the disturbance:

Hs�k� �2 sin�kΔx2

Δx d

dx �sin�kx��x�0

� 2 sinkΔx2

kΔx

: (17)

This is a sinc function, which has zeros for k � 2nπΔx for integers

n. This will only be true for strictly two-dimensional disturb-ances, which is not physical. Fulghum [12] has shown byMonte Carlo simulation that for randomly oriented disturb-ances the transfer function is not oscillatory and does not con-tain zeros. The precise form of the transfer function can intheory be determined by similar means using the FLDI soft-ware presented here, but the process is very time-consuming

Wavenumber [1/mm]10-2 10-1 100 101 102 103 104

Tra

nsfe

r F

unct

ion

Mag

nitu

de [d

B]

-35

-30

-25

-20

-15

-10

-5

0

5

Analytical ResultComputation

Fig. 5. Transfer functionHw�k� for a single beam for 1D sinusoidaldisturbances in x in an infinitesimally thin plane at z � 0.

Wavenumber [1/mm]10-2 10-1 100 101 102 103 104

Tra

nsfe

r F

unct

ion

Mag

nitu

de [d

B]

-35

-30

-25

-20

-15

-10

-5

0

5

L=0 mmL=1 mmL=5 mmL=10 mmL=30 mmL=50 mmL=100 mmL=150 mm

L increasing

Fig. 6. Transfer function of Eq. (16) plotted for various values of L.As L increases or as more signal away from the best focus is considered,the error function in Eq. (16) contributes a k−1 roll-off beginning atlower values of k. This leads to attenuation of high-wavenumber dis-turbances away from the best focus.


and it has been observed that the effect from Hw�k� is muchmore significant.

Hw�k� and Hs�k� are combined for strictly two-dimensional disturbances here for verification purposes becausean overall transfer function can be derived analytically. Becausethe transfer functions are written in wavenumber space, theoverall transfer function H �k� is simply the product of Hw�k�and Hs�k�. For instance, for density disturbances in the infini-tesimal plane at the best focus, the overall transfer function is

H �k� � 2

kΔxsin

�kΔx2

�exp

�−w20k

2

8

�: (18)

Three overall transfer functions for L � 0 mm (infinitesi-mal plane at the best focus), L � 10 mm and L � 30 mm,respectively, are shown in Figs. 9–11. Excellent agreement isobserved between the FLDI software (points in Figs. 9–11)and the analytical functions (lines in Figs. 9–11), affirmingthe accuracy of the computational method. The apparent os-cillations in the transfer functions result from the sinc filterbecause of the strictly two-dimensional nature of the disturb-ance field simulated. It is also worth noting here that there is nofiltering effect resulting from the overlap of the beams awayfrom the best focus as can be seen in Fig. 3. In other words,

the FLDI does not reject signals by “common-mode” rejectionbecause the beams overlap away from the best focus, but be-cause the beam diameter is large compared to the wavelengthsof the disturbances being measured. The fact that the beamsoverlap in space is irrelevant to signal rejection.

Wavenumber [1/mm]10-2 10-1 100 101 102 103

Tra

nsfe

r F

unct

ion

Mag

nitu

de [d

B]

-35

-30

-25

-20

-15

-10

-5

0

5


Fig. 7. Transfer function Hw�k� for a single beam for uniform 2Dsinusoidal disturbances in x between z � �10 mm centered at z � 0.

Wavenumber [1/mm]10-2 10-1 100 101 102 103

Tra

nsfe

r F

unct

ion

Mag

nitu

de [d

B]

-35

-30

-25

-20

-15

-10

-5

0

5


Fig. 8. Transfer function Hw�k� for a single beam for uniform 2Dsinusoidal disturbances in x between z � �30 mm centered at z � 0.

Wavenumber [1/mm]10-2 10-1 100 101 102 103

Tra

nsfe

r F

unct

ion

Mag

nitu

de [d

B]

-35

-30

-25

-20

-15

-10

-5

0

5


Fig. 9. Transfer function H �k� for the two-beam FLDI for 1Dsinusoidal disturbances in x in an infinitesimally thin plane at z � 0.

Wavenumber [1/mm]10-2 10-1 100 101 102 103

Tra

nsfe

r F

unct

ion

Mag

nitu

de [d

B]

-35

-30

-25

-20

-15

-10

-5

0

5


Fig. 10. Transfer function H �k� for the two-beam FLDI for uni-form 2D sinusoidal disturbances in x between z � �10 mm centeredat z � 0.

Wavenumber [1/mm]10-2 10-1 100 101 102 103

Tra

nsfe

r F

unct

ion

Mag

nitu

de [d

B]

-35

-30

-25

-20

-15

-10

-5

0

5


Fig. 11. Transfer function H �k� for the two-beam FLDI for uni-form 2D sinusoidal disturbances in x between z � �30 mm centeredat z � 0.


A convergence study was performed by computing the totalsum-of-squares error for the transfer function shown in Fig. 11as a function of the resolution in z, the direction of mean beampropagation. This is shown in Fig. 12 with the red square in-dicating the chosen resolution. Absolute error is chosen overrelative error because the small absolute errors at small valuesof the transfer function at high wavenumbers are not importantto the output of the software, but these dominate the relativeerror. The grid is converged with approximately 4000 points inz, corresponding to a step size of 257.5 μm. The chosen stepsize of 100 μm is therefore sufficiently small.

The issue of the electronic noise floor can best be seen here.Figure 13 shows the absolute response of the instrument for adisturbance propagating between �10 mm on either side ofthe focus in terms of intensity due to a difference in phasechange, which is converted to voltage by the photodetector.The density field is the same as that used to compute the trans-fer function of Fig. 10. Although the FLDI can accurately mea-sure the derivative of a density field at wavenumbers below theroll-off, the magnitude of the output signal decreases on eitherside of the roll-off point. Evaluating the magnitude of densitydisturbances from an FLDI signal necessarily involves integra-tion to counteract the differentiation performed by the instru-ment. This resolves the issue of the output signal being

lower for low wavenumbers, but one must be aware of thisissue in order to avoid the electronic noise floor of the physicalFLDI system.

B. Comparison with ExperimentAn experiment with a controlled density gradient was devisedto compare the FLDI software against experimental data. Agravitationally stabilized argon jet with a high aspect ratio, rec-tangular cross section is probed with the beams of the FLDIexperimentally, and a model of the resulting density field is in-put into the FLDI software so that the results can be compared.The experimental apparatus is shown in Fig. 14. The primarycomponent is a rectangular cavity of length 165 mm, width10 mm, and depth 152 mm. Argon is fed through the holeat the bottom of the chamber at a specified pressure using aneedle valve. The chamber is filled with 20–40 mesh size (ap-proximately 400–800 μm diameter) corn cob abrasive media toensure the flow at the exit of the chamber is laminar and uni-form across the exit plane. The top of the chamber is coveredwith a woven-wire steel cloth with 230-by-230 μm openings tocontain the abrasive media.

The jet was imaged using schlieren visualization and isfound to achieve the stable configuration shown in Fig. 15. Theargon is moving vertically upward at the chamber exit with anaverage velocity of 0.22 m/s, computed from the measured flowrate delivered by a King Instruments rotameter, but stopsand reverses direction due to gravity. The maximum heightachieved at y � 0 is determined by the flow rate and is typicallyabout 15 mm. A two-dimensional planar computation was per-formed using OpenFOAM with the rhoReactingFoam solver[18]. Figure 16 shows the steady-state result of the computa-tion, with velocity vectors on the left and streamlines on theright with both superimposed on a contour plot of argon massfraction. Numerical schlieren from the computation exhibitsqualitative agreement with the schlieren image in Fig. 15.

The flow is observed to be uniform along the length of thechamber in the z-direction, stable in time, and laminar. Thechamber is placed in the test section of the Caltech LudwiegTube and made parallel to the z-axis of the FLDI beams byusing a level suspended between two parallel circular cavities.

Step size [mm]10-2 10-1 100 101

Err

or

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

Fig. 12. Convergence study for the transfer function shown inFig. 11. The red square indicates the chosen resolution.

Wavenumber [1/mm]10-2 10-1 100 101 102 103 104O

utpu

t sig

nal m

agni

tude

[arb

. uni

ts, n

orm

aliz

ed]

10-5

10-4

10-3

10-2

10-1

100

Fig. 13. Normalized FLDI output signal for sinusoidal disturbanceswithin�10 mm of the focus in z, corresponding to the transfer func-tion plotted in Fig. 10.

Fig. 14. Solid model of the chamber for the argon jet with dimen-sions given in mm. The coordinate system shown corresponds to theorientation of the coordinate system of the FLDI beams.


The FLDI beams are separated in the x-direction andlocated at the interface of the argon and air, such that theymeasure the density change across the interface as shown inFig. 15(a). The resulting phase change is less than π∕2,so the small-angle approximation can be invoked as inSection 2. Portions of the jet can be covered with tape suchthat only sections of the jet are active. Fluctuations in the am-bient air are negligible compared to the density differenceacross the interface. The optical index field can therefore beapproximated for computational purposes as varying in xand being uniform in y and z over the portion(s) where thechamber exit is not covered and uniform everywhere else,i.e., n 0 � njet�x��U �z0 � L∕2� − U �z0 − L∕2��. This allowsfor comparison between the FLDI software and experiment fora number of different configurations. At each condition, theflow rate of the argon is adjusted to produce the maximumFLDI signal, meaning that the beams are centered on themaximum of the density gradient. Test cases performed arepresented in Table 2 by total length of the jet in z (Ljet)and the center of the jet in z (z0) (units in mm).

The variation in index of refraction in the x-direction acrossthe argon–air interface is shown in Fig. 17. Cubic spline inter-polation is used in the FLDI software when evaluating the in-dex of refraction on the computational grid. The index ofrefraction for a mixture of gases is calculated from Eq. (19),

which is derived from Eq. (B29) in [19] by substituting thedefinition of mass fraction �Y �:

n 0 � Y Ar�ρAr � ρair��K Ar −K air��K air�ρAr � ρair − ρairjx→∞�:(19)

The computational and experimental FLDI phase changeoutputs for each case in Table 2 are plotted versus one anotherin Fig. 18. Uncertainty in the experiment is difficult toquantify, but errors are believed to be largely due to three-dimensional effects at the ends of the jet. The error bars inFig. 18 are a bound on the error from calculating the responsewith an additional 5 mm of jet length on either side of the jetfor each configuration. Three-dimensional effects have a largerinfluence on shorter jet lengths, which explains the scatter inthe data at the low-response end of the figure. A linear regres-sion analysis was carried out to test the correlation of experi-mental data and numerical results. The regression line has aslope of 0.99 and an intercept of 0.00, compared to the idealvalues of 1 and 0 for slope and intercept, respectively, and allbut one of the individual points lie on the regression line withinthe computed uncertainty. The FLDI software can therefore beconsidered verified versus analytical calculations and validatedagainst experimental data with a high degree of confidence.

Fig. 15. (a) Schlieren image of the argon jet in the x–y plane. Thetop of the jet is about 15 mm from the chamber exit and is observed tobe uniform across the jet cross section, temporally stable, and laminar.(b) Pseudo-schlieren image of the argon jet showing the vertical gra-dient in density for comparison with the experiment.

Fig. 16. Steady-state result of the OpenFOAM computation of theflow out of the argon jet. Velocity vectors are shown on the leftand streamlines on the right. Both are superimposed on a contour plotof argon mass fraction. Ljet is the length out of the page (in thez-direction) that the cross section shown here extends.

Table 2. Argon Jet Configurations Tested

Configuration Ljet z0A 165 0B 130 0C 110 0D 90 0E 70 0F 50 0G 30 0H 20 0I 10 0J 82.5 41.25K 20 10L 20 20M 20 30N 32.5 66.25

x [mm]-20 -10 0 10 20

Δ n

× 10-6

-8

-7

-6

-5

-4

-3

-2

-1

0

OpenFOAM dataSpline fit

Fig. 17. Index of refraction field from the OpenFOAM computa-tion with spline interpolation.


5. SIMULATED MEASUREMENTS

A. DescriptionOne of the more promising applications of the FLDI is meas-uring instability waves in hypervelocity boundary layers whereother more conventional techniques such as surface-mountedpressure transducers or hot-wire anemometry are not suitable.The FLDI is capable of making such measurements because ofits high temporal resolution and ability to reject signals awayfrom the best focus. This was done by Parziale at Caltech in hisPh.D. thesis [11]. These instability waves are Mack-modewaves and propagate two-dimensionally in the mean flowdirection. Because the waves are two-dimensional, one can as-sume that waves only propagate along the x-axis of the FLDI.As such, the FLDI is quite capable of measuring their fre-quency, but because of the path-integrated nature of FLDImeasurements it is impossible to determine their amplitudefrom the FLDI output alone. This point is possibly made clear-est by Smeets himself [20]: “From the signals received [from a2D shear layer], primarily qualitative information could beachieved, e.g., on the frequency spectra of the turbulent fluc-tuations near the focal point. The deduction of quantitativedata on the level of local density fluctuations is only possibleby means of assumptions and approximations. The accuracy ofthe results is, therefore, only moderate.”

From Eq. (6) we see that the change in phase between thetwo FLDI beams, which is converted to a voltage by the photo-detector, is an integrated function of the density fluctuationsalong the beam paths as

ΔΦ�t�Δx

� 2πKλΔx

Z ZDI0�ξ; η�

×�Z

D�ξ;η�

s1ρ 0�x1; t�ds1 −

ZD�ξ;η�

s2ρ 0�x2; t�ds2

�dξdη;

(20)

where ρ 0 represents the local density fluctuations. We seek theamplitude of ρ 0, the density fluctuation. If Δx is sufficientlysmall and recalling that x1 � x2 � Δxx, we can write thisequation in terms of derivatives in x instead of finite differences

and approximate the two beam paths in Eq. (20) as beingcommon:

s1 ≈ s2 ≈ s; (21)

ρ�x1� ≈ ρ�x2� �∂ρ∂x

Δx; (22)

dΦ�t�dx

≈2πKλ

Z ZDI 0�ξ; η�

Zs

∂ρ 0�x; t�∂x

dsdξdη: (23)

Since the output of the FLDI is a function of time, not space, itis much more useful to take derivatives with respect to time.This conversion can be done using Taylor’s hypothesisx � cr t for a constant phase speed cr . We then have

dΦ�t�dt

≈2πKλ

Z ZDI 0�ξ; η�

Zs

∂ρ 0�x; t�∂t

dsdξdη: (24)

We can then apply the mean value theorem for integrals to theright-hand side to obtain

dΦdt

� 2πKλ

Z∂ρ 0

∂t: (25)

Here, ∂ρ0

∂t represents the averaged value of the time derivative ofρ 0 over the spatial integral. Z is an unknown parameter withunits of length that makes ρ 0 equal to the actual ρ 0 fluctuationin the boundary layer. It is approximately equal to the length ofthe region where the FLDI is probing the boundary layer, but itcannot be determined a priori from experimental data alone.However, Z is primarily a function of the flow geometryand can be calculated with knowledge from the FLDI softwarefor a given geometry and used with experimental data for thesame flow. It is sensitive to the spatial filtering of the FLDI asindicated in Eq. (16), so care must be taken to ensure that thewavelengths being measured are not significantly attenuated.

We can then integrate the FLDI output in time to obtainthe density fluctuations as a function of time,

crΔx

Zt

0

ΔΦ�τ�dτ � 2πKλ

Zρ 0�t�; (26)

or, solving for eρ 0,

ρ 0�t� � crλ2πΔxZK

Zt

0

ΔΦ�τ�dτ: (27)

The integral can be evaluated using standard numerical integra-tion methods, e.g., Simpson’s rule. Alternatively, since instabil-ity waves are often analyzed in frequency space, we can writethe Fourier transform in time of Eq. (27) as

F �ρ 0� � crλ2πΔxZK

1

iωF �ΔΦ�: (28)

The FLDI software can be used to make simulated measure-ments of a Mack-mode wave packet in a boundary layer, andthe result of that simulation can be used to calculate a value forZ that can be used in experiments to determine the magnitudeof the fluctuations. The wave packet is calculated for T5shot number 2789. The boundary-layer edge conditions areas follows.

The FLDI measurement location is 710 mm from the tip ofa 5° half-angle cone model as shown in Fig. 19. Wave packet

Fig. 18. Experimental versus numerical data for the argon jet ex-periments detailed in Table 2. The letters marking each data pointcorrespond to the configurations in Table 2. The line is the ideal liney � x.


characteristics are calculated using the method described in [21]and [22].

The density field in a hypersonic boundary layer contains apropagating disturbance that can be approximated as

ρ�x; y; t� � ρ�x; y� � ρ 0�x; y; t�; (29)

with a mean spatial density field ρ�x; y� and a fluctuatingcomponent ρ 0 representing a wave packet containing high-frequency waves representative of Mack (second-mode) insta-bilities, where

ρ 0 � ρSFE�x; t�ρ 0: (30)

Here, ρSF is a dimensional scale factor that determines the am-plitude of the density fluctuation. It is chosen such that themaximum density fluctuation is 0.1% of the freestream value.E�x; t� is an envelope describing the extent of the wave packetin x, and ρ 0 is a nondimensional fluctuation of the form

ρ 0 � Re�g�y� exp�i�αx − ωt��; (31)

where g is a complex eigenfunction in y, α is a complex wave-number containing information on both the spatial wavenum-ber (αr � k) and the spatial growth rate (αi), and ω is thetemporal frequency of the wave. Written in terms of realand imaginary parts, ρ 0 is given by

ρ 0 � exp�−αix��gr�y� cos�αrx − ωt� − gi�y� sin�αrx − ωt��:(32)

The eigenfunctions and eigenvalues were computed by Bitter[21] using parallel flow linear stability theory for a boundarylayer on a cone with flow conditions given in Table 3. The realand imaginary eigenfunctions are plotted versus height abovethe cone surface in Fig. 20 along with the boundary layer.

From Fig. 20(c) and examination of the cone geometry it isdetermined that the FLDI beams are inside the boundary layerwithin about 10 mm on either side of the best focus. Theenvelope is chosen to be a Gaussian that contains about 15wavelengths inside the region where its value is greater than1%. Functionally, this has the form

E�x; t� � exp

�−A�x − cr t − x0�2

l2

�; (33)

for envelope length l and spatial starting location x0.Equation (30) has the final form of

ρ 0 � ρSF exp

�−A�x − cr t − X 0�2

l2− αix

��gr�y� cos�αrx − ωt�

− gi�y� sin�αrx − ωt��: (34)

The parameters in Eq. (34) have the values given in Table 4. Itis important to recognize that we should not expect significantspatial filtering for αr � 2.116∕mm based on Fig. 10. Notethat Fig. 10 does not represent the transfer function for thiscone boundary-layer flow but is the transfer function fortwo-dimensional disturbances propagating in x and uniformin z from −10 < z < 10 mm, so the exact magnitude of thetransfer function at a given wavenumber is not directly appli-cable here. Still, Fig. 10 does predict whether or not a particularwavenumber will be appreciably attenuated.

The FLDI beams are positioned in the simulation at thelocal maximum of the density eigenfunction at y � 0.81 mm.The density fluctuations are converted to refractive indexfluctuations by the Gladstone–Dale relation [Eq. (7)] withK � 0.227 × 10−3 m3∕kg. The wave packet is assumed tobe axisymmetric with respect to the cone axis. The local density

Fig. 19. Cone model used in the T5 studies.

Table 3. Boundary-Layer Edge Conditions for the T5Shot 2789

Me 4.55T e 2105 KUe 4191 m/spe 47.1 kPaRe/m 4.76 × 106∕mρe 0.0777 kg∕m3

Eigenfunction magnitude [arbitrary units]

-2 0 2 4 6 8 10

y [m

m]

0

0.5

1

1.5

2

2.5

3

Eigenfunction magnitude [arbitrary units]

-5 0 5 10

y [m

m]

0

0.5

1

1.5

2

2.5

3

Density [kg/m3]

0 0.2 0.4 0.6

y [m

m]

0

0.5

1

1.5

2

2.5

3

(a) (b)

(c)

Fig. 20. (a) Real eigenfunction. (b) Imaginary eigenfunction.(c) Mean density profile ρ�y�.

Table 4. Boundary-Layer Wave Packet Parameters

ρSF 2.276 × 10−20 kg∕m3

A − ln 0.01 � 4.6L 14π∕αr � 20.8 mmx0 710 mm − l � 689.2 mmcr 3633 m/sαi −0.0488/mmαr � k 2.116/mmω � 2πf � crαr 7.697 × 106 rad∕s


is transformed from the FLDI coordinate system (subscript f )to a coordinate system relative to the cone surface normal(subscript c), so that Eq. (34) can be evaluated by a seriesof trigonometric operations for cone half-angle θc . Note thatat zf � 0, �xf ; yf � � �xc; yc� as expected.

yc � cos θc

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi��xf − yf tan θc� tan θc �

yfcos θc

�2

� z2f

s

− tan θc�xf − yf tan θc��; (35)

xc � xf � tan θc�yc − yf �: (36)

The index of refraction change, relative to the index of refrac-tion outside the boundary layer, at a point �xc; yc ; t� is

Δn � K �ρ�yc� � ρ 0�xc; yc ; t� − ρe�: (37)

Functions ρ, gr , and gi are tabulated on a uniform stencil in ycwith a step size of 0.74 μm, and values are interpolated from thetabulated functions for each simulated ray in the FLDI soft-ware. The wave packet is propagated in time and the samplerate of the simulated FLDI is 20 MHz, which is sufficient tofully temporally resolve the wave packet.

B. ResultsThe output of the simulation is shown in Fig. 21 along with theinput density disturbance at the probe location in x, which ishalfway between the two FLDI beams and z � 0, i.e., the bestfocus. Note that the output is in phase change, as it would be ifoutput from a photodetector in a physical FLDI setup, and thatthe output is proportional to the time derivative of the inputwave packet as indicated by Eq. (27).

Equation (27) is applied to the simulated FLDI output andZ is determined by means of iteration until the peak magnitudeof the measured power spectral density of the simulated outputis equal to that of the input. For the flow geometry studiedhere, Z is calculated to be 12.6 mm. Figure 22 shows boththe measured and input density fluctuations with respect totime. Note that the only difference between the two is that

the measured fluctuations are slightly ahead of the input withrespect to time. This is simply a consequence of the conicalgeometry of the flow: the wave packet will first be observedby the FLDI beams away from the focus. Equations (35) and(36) make the issue clear. For z ≠ 0, yc > yf so xc > xf .

Figure 23 shows the power spectral density (PSD) of theinput and output density fluctuations. The two curves matcheach other almost exactly with the appropriate value of Z exceptthat the simulated measurement has higher noise away from thepeak in frequency. Presumably even more noise would bepresent in an experiment, so this slight increase in noise isinconsequential.

The parameters in Table 4, namely, αr and cr , are changedto determine the universality of the value of Z for wave packetsof different spatial wavenumber and frequency. The values forαr and cr are given in Table 5 along with the computed fre-quency and the error in the peak amplitude of the simulatedmeasurement relative to the peak magnitude of the input wavepacket.

The error in the peak amplitude of the wave packet is ap-preciable only in cases 3 and 4. The reason for the large error

Fig. 21. Output of the FLDI simulation compared to the inputboundary-layer wave packet density disturbance. The y-axis on the leftcorresponds to the FLDI output and the y-axis on the right corre-sponds to the input density at the location of the beams at z � 0.

Fig. 22. Output of the FLDI simulation, scaled with Eq. (27), com-pared to the input boundary-layer wave packet for the optimum valueof Z � 12.6 mm.

Fig. 23. PSD of the FLDI simulation, scaled with Eq. (27) andoptimum Z value, compared to the PSD of the input boundary-layerwave packet.


can be deduced by examining the values of αr in these cases andconsulting Fig. 10. Recall again that the transfer functions plot-ted in Figs. 9–11 are not transfer functions for the conicalboundary layer under examination in this section, but nonethe-less they lend some insight to when the instrument will begin toroll off in wavenumber space. Notice that at αr � 7∕mm, thetransfer function has dropped to about −2 dB, which corre-sponds to about a 35% reduction in signal magnitude, closeto the observed error in Table 5. Similarly, for αr � 30∕mm,the transfer function is approximately equal to −17 dB, corre-sponding to a 98% reduction in magnitude, which is againquite close to the error in Table 5 for case 4. Physically, thismeans that the wavelength of the Mack-mode waves is suffi-ciently small enough that it is comparable in size to the beamwidth in the region where the FLDI is probing the boundarylayer. Therefore, the waves are being heavily spatially filtered bythe beams.

It is interesting to examine further the output from theFLDI software for case 4 where αr � 30∕mm. The densitychange as a function of time and the power spectral densityof the disturbance are shown in Fig. 24. As Table 5 indicates,the measured signal magnitude is significantly reduced com-pared to the input disturbance using the same value of Z cal-culated for case 1 where there is very little spatial filtering by theinstrument. Interestingly, though, the FLDI is able to measurethe frequency of the signal correctly. This is a somewhat sur-prising result because if one were to approximate the FLDI astwo point measurements separated by the beam separation Δx,one would predict spatial aliasing if the wavenumber of the dis-turbance is greater than π

Δx, which would be 18 mm here. Thisspatial aliasing would become temporal aliasing when Taylor’s

hypothesis is applied. Figure 24(b) clearly does not display thisbehavior, so the finite beam diameter seems to prevent spatialaliasing based on wavenumber. An FLDI will only suffer fromaliasing if the sampling frequency is not sufficiently high com-pared to the frequency of the disturbance being measured. Theonly effect of high wavenumbers is the attenuation of the signaldue to spatial filtering, which will lead to a limitation from theelectronic noise floor of the system.

Based on this analysis we conclude that the value of Z inEq. (27) is only a function of the geometry of the density fieldbeing probed and the attenuation of the signal due to spatialfiltering based on wavenumber. If the attenuation is sufficientlysmall for all wavenumbers of interest in the experiment thenonly a single value of Z needs to be calculated to accurately com-pute the density change quantitatively. Even if the wavenumbersof interest are high enough to be filtered by the FLDI, new valuesof Z can be calculated for given wavenumbers without muchdifficulty. The measurement will still be a point measurementat the focus of the FLDI and the phase speed of the disturbancemust be known a priori, but for the specific application ofmeasuring Mack-mode waves on a slender-body boundary layerthese restrictions are not problematic. The phase speed caneither be calculated from stability theory as is done here or itcan be measured as in [23]. One also needs some knowledgeregarding the shape of the density eigenfunctions.

To summarize, the routine to accurately compute the den-sity change of Mack-mode waves in a hypervelocity boundarylayer from FLDI data is as follows:

1. Compute wave properties and eigenfunctions fromlinear stability analysis, e.g., in [21].

2. Input this data into FLDI software such as the one inthe current study.

3. Compute a value for Z in Eq. (27) by matching theresults of step 2 to the specified density change of the inputwave packet.

4. Apply Eq. (27) to the experimental FLDI data using thevalue of Z from step 3, taking care to ensure that the wave-number of the Mack-mode wave is not appreciably spatiallyfiltered by the FLDI.

The simple procedure used in [11] for computing the den-sity disturbance magnitude for the T5 shot 2789 can give anestimate accurate within an order of magnitude if the “integra-tion length” is estimated based on the length through which theFLDI beams are inside the cone boundary layer. However, sucha procedure makes significant approximations that limit itsaccuracy to within about a factor of 2. The routine outlinedabove, on the other hand, requires knowledge of the formof the disturbances and is limited in accuracy by experimentaluncertainty and any uncertainties associated with the compu-tations performed in step 1. One could alternatively simplymake an estimate of Z based on the boundary-layer geometryand skip directly to step 4 above. Estimating Z would likelygive results that are more accurate than the procedure in [11]because Eq. (27) takes into account the fact that the FLDI isdifferentiating the density disturbance in space while the sim-plified procedure does not. However, the results would be lessaccurate than those obtained by following the full routineoutlined above.

Table 5. Wave Packet Properties Tested with FixedValue of Z

Case No. αr [1/mm] cr [m/s] f [Hz] % Error

1 2.116 3633 1.22 × 106 0.02 0.4 3633 2.31 × 105 4.03 7 3633 4.05 × 106 −264 30 3633 1.73 × 107 −945 2.116 2500 8.42 × 105 0.06 2.116 5000 1.68 × 106 0.0

(a) (b)

Fig. 24. (a) Output of the FLDI simulation for case 4 in Table 5scaled with the same value of Z found in case 1 compared to the actualdensity disturbance at the focus. (b) Power spectral density of theFLDI simulation for case 4 compared to the PSD of the inputboundary-layer wave packet at the focus.


There is a caveat to these results. The freestream disturb-ances outside the boundary layer have been neglected in thisanalysis, but in an experiment the FLDI beams integratethrough these disturbances and they will contribute to the out-put signal. Therefore, care must be taken to ensure that thefreestream disturbances do not occupy the same frequencyspace as the disturbance of interest at the focus or that the wave-number of the freestream disturbance at the frequency of thedisturbance of interest is sufficiently high that it is heavily spa-tially filtered by the FLDI away from the focus. It is importantto recognize that the characteristic velocity of the freestreamdisturbances can be different from the characteristic velocityof the disturbance of interest, as it is for hypervelocity boundarylayers. Based on measurements taken in T5, it appears that thefrequency range over which there is significant FLDI signal inthe freestream is significantly below the 1.2 MHz of a typicalMack-mode wave packet, so signal contamination from thefreestream is not expected to be problematic when using theFLDI to measure Mack-mode waves on a cone boundary layer.The effect of freestream disturbances on the output signalought to be evaluated for each facility on a case-by-case basisfor future experiments before attempting to quantify boundary-layer instability measurements to ensure the measurements areaccurate. This is particularly true for low-enthalpy facilitieswhere the frequency content of the freestream disturbancesand that of Mack-mode waves are not as far apart as theyare in high-enthalpy flow.

Another word of caution is warranted regarding measuringMack-mode waves in hypervelocity boundary layers with anFLDI. From Fig. 20, it is clear that the magnitude of densityfluctuations varies significantly with height in the boundarylayer, and as such the output of the FLDI is quite sensitiveto the location above the cone surface where the disturbanceis measured. Indeed, if the measurement location is moved only400 μm from the specified measurement location of y �0.81 mm to y � 1.2 mm, the error in the measurement usingthe value of Z calculated for case 1 in Table 5 is greater than50%. Therefore, great care must be taken to accurately measurethe height of the beam centers from the model surface in anexperiment.

6. CONCLUSIONS

Focused laser differential interferometry is a promising tech-nique for measuring localized, high-frequency density disturb-ances in supersonic and hypersonic flows. In particular, theFLDI is an attractive instrument for measuring Mack-mode in-stability waves in hypersonic boundary layers. A computationaltool has been developed here that has been verified against ana-lytical predictions of the FLDI response as well as experimentalmeasurements with a physical FLDI setup. Using the FLDIsoftware, it is possible to calculate the output of an FLDI toan arbitrary density field. Such a prediction can only be madeanalytically for very simple density fields. These computationsallow FLDI experiments to give quantitative measurements ofthe density fluctuation amplitude if certain details about theflow in question are known by applying correction factors tothe computational output such that the FLDI output matchesthe input density disturbance of interest. This procedure is

shown to work well for hypersonic boundary-layer disturbanceswhere the disturbance is localized near the best focus of theFLDI, is two-dimensional in nature, and the phase speed ofthe instabilities can be accurately predicted from theory.

Some general statements can be made concerning using theFLDI to make measurements in compressible flows. The FLDIdoes reject signals away from its best focus, but this signal re-jection is not related to common-mode rejection associatedwith the beams sharing common paths. The rejection is in factattenuation due to spatial filtering because of the increasingbeam diameters away from the focus. As such, disturbanceswith small enough wavenumbers (long enough wavelengths)will not be attenuated away from the FLDI focus and willtherefore contribute to the FLDI signal over a significant extentof the beam paths. As a result, the FLDI can accurately measurethe frequency content of density disturbances in a flow, but itdoes not in general yield information as to where along thebeam path the disturbances are located or their amplitude atany given point along the beam path. In order to extract quan-titative density information, details regarding the geometry ofthe density disturbance field, the preferred direction of the dis-turbances (if any), and the characteristic velocity of the disturb-ances must be known. If these are known, then the FLDIoutput can be simulated using a procedure such as the one pre-sented here or, if the flowfield is simple enough, by analyticalmethods, and then experimental measurements can be adjustedas necessary such that the density fluctuation magnitude is cor-rect. The authors are not aware of any other method to extractsuitably accurate quantitative density fluctuation magnitudesfrom FLDI data.

Acknowledgment. The authors would like to thankDr. Matthew Fulghum for helpful insights into how theFLDI attenuates unwanted signals. The authors would also liketo thank Aditya Bhattaru for his assistance setting up the FLDIat Caltech during his Summer Undergraduate ResearchFellowship in 2014. The authors would additionally like tothank Dr. Josué Melguizo-Gavilanes for performing theOpenFOAM computations in Section 4.B. Finally, the authorsare very grateful to Neal Bitter for performing the boundary-layer stability calculations that provided the data for Section 5.The lead author acknowledges the support of Foster and CocoStanback STEM Fellowship for his graduate studies.

REFERENCES1. G. Smeets, “Observational techniques related to differential

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